Representation of linearly additive random fields (Q1184047)

Chentsov type representation theorem is proved for stochastically continuous, linearly additive, infinitely divisible random field without Gaussian component, where a random field \(X=\{X(t),\;t\in R^ d\}\) is called linearly additive if the stochastic process \(\xi\) defined by \(\xi(\lambda)=X(a+\lambda b)\), \(\lambda\in R\), has independent increments for every pair \((a,b)\), \(a,b\in R^ d\). In passing it is shown that there exists a natural one-to-one correspondence between stochastically continuous, linearly additive Poisson random fields on \(R^ d\) and locally finite, bundleless measures on the space of all \((d-1)\)- hyperplanes in \(R^ d\). The latter result is closely related to Ambartzumian's theorem on the representation of linearly additive pseudo- metrics in the plane.